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    <H1>Chicago undergraduate mathematics bibliography</H1>

<P>Somehow I became the canonical undergraduate source for
bibliographical references, so I thought I would leave a list behind
before I graduated.  I list the books I have found useful in my
wanderings through mathematics (in a few cases, those I found
especially <i>un</i>useful), and give short descriptions and
comparisons within each category.  I hope that this list may serve as
a useful &ldquo;road map&rdquo; to other undergraduates picking their
way through Eckhart Library.  In the end, of course, you must explore
on your own; but the list may save you a few days wasted reading books
at the wrong level or with the wrong emphasis.

<P>The list is biased in two senses.  One, it is light on foundations
and applied areas, and heavy (especially in the advanced section) on
geometry and topology; this is a consequence of my interests.  I
welcome additions from people interested in other fields.  Two, and
more seriously, I am an honors-track student and the list reflects
that.  I don't list any &ldquo;regular&rdquo; analysis or algebra
texts, for instance, because I really dislike the ones I've seen.  If
you are a 203 student looking for an alternative to the awful pink
book (Marsden/Hoffman), you will find a few here; they are all much
clearer, better books, but none are nearly as gentle.  I know that
banging one's head against a more difficult text is not a realistic
option for most students in this position.  On the other hand, reading
mathematics can't be taught, and it has to be learned sometime.  Maybe
it's better to get used to frustration as a way of life sooner, rather
than later.  I don't know.

<P>Reviews not marked with initials, or marked with <b>[CJ]</b>, were
written by me, Chris Jeris ('98).  Other contributors are marked:
<b>[PC]</b>, Pete Clark ('98); <b>[PS]</b>, Pete Storm ('98);
<b>[BB]</b>, Ben Blander ('98); <b>[RV]</b>, Rebecca Virnig ('00);
<b>[BR]</b>, Ben Recht ('00); <b>[MG]</b>, Marci Gambrell ('99);
<b>[YU]</b>, Yuka Umemoto ('97).  Thanks to all of them for their
input.

<P>Jump to the <A HREF="#elementary">elementary</A>, <A
HREF="#intermediate">intermediate</A>, or <A
HREF="#advanced">advanced</A> sections.

<P>Warning: Statements about books I haven't looked at in a couple of
years may be factually incorrect; please forgive my spotty memory.  I
don't think I have any really egregious falsehoods in here.  I
apologize for the appearance of this page; most web browsers have not
yet been updated to handle the HTML4 entity set, so fools like me who
read the definition write ugly-looking pages.

<P>Enough apologia.  Here we go:

<P><HR><A NAME="elementary"><H1>ELEMENTARY</H1></A>
This includes &ldquo;high school topics&rdquo; and first-year calculus.

<H3>Contents</H3>
<UL>
  <LI><A HREF="#e:algebra">Algebra</A> (4)
  <LI><A HREF="#e:geometry">Geometry</A> (2)
  <LI><A HREF="#e:foundations">Foundations</A> (1)
  <LI><A HREF="#e:problem-solving">Problem solving</A> (4)
  <LI><A HREF="#e:calculus">Calculus</A> (6)
  <LI><A HREF="#e:bridges">Bridges to intermediate topics</A> (2)
</UL><HR>

<A NAME="e:algebra"><H2>Algebra</H2></A>

<H4>Gelfand/Shen, <i>Algebra</i></H4>
<H4>Gelfand/Glagoleva/Shnol, <i>Functions and graphs</i></H4>
<H4>Gelfand/Glagoleva/Kirillov, <i>The method of coordinates</i></H4>

<P>These three little white books come from the Soviet correspondence
school in mathematics, run by I. M. Gelfand for interested people of
all ages in the further reaches of the USSR.  Rather than trying to be
artificially &ldquo;down-to-earth&rdquo; in the way Americans do,
Gelfand simply assumes that you can understand the mathematics as it's
done (and avoids the formal complexities mathematicians are inured
to).  YSP and SESAME give these out by the carload to their students,
who mostly love them.  <i>TMoC</i> is notable for its intriguing
four-axis scheme for making flat graphs of <b>R</b><sup>4</sup>.  Overall a
fresh, inspiring look at topics we take for granted, and a good thing
to recommend to bright younger students or friends (or parents!)

<H4>Cohen, <i>Precalculus with unit circle trigonometry</i></H4>

<P><b>[RV]</b> I used this book in high school and absolutely loved
it. It's very skimpy on proofs, and really should not be used for that
sort of insight.  However, in terms of understanding how to apply
various mathematical concepts it's wonderful.  It has a large number
of graphs, examples, and easy reference tables. It covers all the
algebra, trig, and cartesian geometry that any good high school math
sequence should deal with. I have used it for years as a reference
book (e.g., what exactly is Cramer's rule again...) Solutions to a
number of the problems are in the back, and the problems are not
entirely applications.

<A NAME="e:geometry"><H2>Geometry</H2></A>

<H4>Euclid, <i>The elements</i> <A href="http://www.ams.org/mathscinet-getitem?mr=17%2C814b">[MR 17,814b]</A></H4>

<P>No, I'm not kidding.  At first it's incredibly annoying and tedious
to read, but after a while you get into the flow of the language and
the style.  Euclid teaches you both the power of the modern algebraic
methods and the things that are hidden by our instinct to assign a
<i>number</i> to a length.  Besides, there are wonderful tidbits here
and there (did you know that Euclid invented the Dedekind cut?).  At
least check it out once, to read his proof of the Pythagorean theorem.
(Thanks to Jonathan Beere ('95) for convincing me it was worthwhile.)

<P><b>[PC]</b> I have Volume I, and I have to admit I haven't really
read it.  I do think that I would benefit if someone rammed some of it
down my throat though, because nowadays we undergraduates are trained
to regard &ldquo;geometric&rdquo; as a strong pejorative&mdash;the very
antithesis of rigor and proof.

<H4>Coxeter, <i>Geometry revisited</i></H4>

<P>This is a text on &ldquo;advanced Euclidean geometry&rdquo;,
starting with the numberless classical &ldquo;centers&rdquo; of a
triangle and proceeding from there.  Many good exercises.  There are
lots of &ldquo;college geometry&rdquo; texts you can find this stuff
in, but most of them are aimed at math-ed majors; this book and
Coxeter's other one (see below) have them all beat.

<P><b>[PC]</b> I like this book.  I don't own it but I've flipped
through it more than once and I agree that it has a pleasantly
non-brain-dead quality to it.  There are interesting geometric facts
that you probably haven't seen before in here.

<A NAME="e:foundations"><H2>Foundations</H2></A>

<H4>Rucker, <i>Infinity and the mind</i></H4>

<P><b>[RV]</b> This is not really a math book.  It is a friendly
introduction to the concept of infinity, transfinite numbers, and
related paradoxes.  I'd recommend it to high school students who are
intrested in math, but not quite ready to sit down and read though
proof after proof of theorems. (In fact, I first read it in high
school as part of an independent study math class.) The book does
contain some proofs, but not in the rigorous form of a standard math
text. It does include more historical background on the concepts than
most math texts do, which is nice. Each chapter is accompanied by
problems, and an answer key (with explanations) is at the end of the
book.

<A NAME="e:problem-solving"><H2>Problem solving (pre-college)</H2></A>

<H4>NML problem books</H4>

<P>The MAA publishes a series called &ldquo;New Mathematical
Library&rdquo; which contains many excellent titles aimed at or below
college sophomore level (<i>Geometry revisited</i> is among them).  In
this series are four books of problems given on the AHSME, one of
USAMO and two of IMO problems, all with solutions.  We use the AHSME
books extensively at YSP; the USAMO and IMO problems still give me a
rough time, and are fun if you're looking for frustration one evening.

<H4>Larson, <i>Problem solving through problems</i></H4>

<P>After you grapple with the IMO problems for a while, turn here to
find a book that teaches (as much as any book can) the art of solving
them.  Cognitive strategies are laid out with examples of problems
(mostly from Olympiads and Putnams) to which they apply.

<P><b>[PC]</b> I own this, or at least I did&mdash;I haven't seen it
since high school.  I'm really <i>not</i> a big contest
problem-solver, but I did use this book and I think it helped to
prepare me for Chicago Mathematics.  Lots of good problems, not all of
them inane.

<H4>P&oacute;lya, <i>How to solve it</i></H4>

<P>I haven't read this, but it's supposed to be the
&ldquo;classic&rdquo; version of Larson above.

<H4>P&oacute;lya, <i>Mathematics and plausible reasoning</i>, I and
II</H4>

<P><b>[PC]</b> These are the &ldquo;sequels&rdquo; to P&oacute;lya's
<i>How to solve it</i>.  They are definitely interesting, although
their main interest may be psychological/philosophical (only relative
to mathematics do philosophy and psychology merge!)  I'm not sure that
one can really become a significantly better problem-solver by reading
a book about the nature of mathematical reasoning, but I admire
P&oacute;lya for writing an interesting and challenging book about the
<i>practice</i> of mathematics; such books are in my opinion too few
and far between.

<P>In 1997&ndash;98 a few books with the same general theme as Larson,
but different problem collections, have been published; I haven't seen
any of them.

<A NAME="e:calculus"><H2>Calculus</H2></A>

<P>Of course, as we all know, the One True Calculus Book is

<H4>Spivak, <i>Calculus</i></H4>

<P>This is a book everyone should read.  If you don't know calculus
and have the time, read it and do all the exercises.  Parts 1 and 2
are where I finally learned what a limit was, after three years of
bad-calculus-book &ldquo;explanations&rdquo;.  The whole thing is the most
coherently envisioned and explained treatment of one-variable calculus
I've seen (you can see throughout that Spivak <i>has a vision</i> of
what he's trying to teach).

<P>The book has flaws, of course.  The exercises get a little
monotonous because Spivak has a few tricks he likes to use repeatedly,
and perhaps too few of them deal with applications (but you can find
that kind of exercise in any book).  Also, he sometimes avoids
sophistication at the expense of clarity, as in the proofs of Three
Hard Theorems in chapter 8 (where a lot of epsilon-pushing takes the
place of the words &ldquo;compact&rdquo; and &ldquo;connected&rdquo;).
Nevertheless, this is the best calculus book overall, and I've seen it
do a wonderful job of brain rectification on many people.

<P><b>[PC]</b> Yes, it's good, although perhaps more of the affection
comes from more advanced students who flip back through it?  Most of
my exposure to this book comes from tutoring and grading for 161, but
I seriously believe that working as many problems as possible (it must
be acknowledged that many of them are difficult for first year
students, and a few of them are <i>really</i> hard!) is invaluable for
developing the mathematical maturity and epsilonic technique that no
math major should be without.

<P>Other calculus books worthy of note, and why:

<H4>Spivak, <i>The hitchhiker's guide to calculus</i></H4>

Just what the title says.  I haven't read it, but a lot of 130s
students love it.

<H4>Hardy, <i>A course of pure mathematics</i> <A href="http://www.ams.org/mathscinet-getitem?mr=94c%3A00002">[MR 94c:00002]</A></H4>
<H4>Courant, <i>Differential and integral calculus</i> <A href="http://www.ams.org/mathscinet-getitem?mr=90e%3A26001a">[MR 90e:26001a-b]</A></H4>

<P>These two are for &ldquo;culture&rdquo;.  They are classic
treatments of the calculus, from back when a math book was rigorous,
period.  Hardy focuses more on conceptual elegance and development
(beginning by building up <b>R</b>).  Courant goes further into
applications than is usual (including as much about Fourier analysis
as you can do without Lebesgue integration).  They're old, and old
books are hard to read, but usually worth it.  (Remember what Abel
said about reading the masters and not the pupils!)

<H4>Apostol, <i>Calculus</i> <A href="http://www.ams.org/mathscinet-getitem?mr=35+%235554">[MR 35 #5554]</A> <A href="http://www.ams.org/mathscinet-getitem?mr=40+%231542">[MR 40 #1542]</A></H4>

<P>This is &ldquo;the other&rdquo; modern rigorous calculus text.
Reads like an upper-level text: lemma-theorem-proof-corollary.  Dry
but comprehensive (the second volume includes multivariable calculus).

<H4>Janusz, <i>Calculus</i></H4>

<P>The worst calculus book ever written.  This was the 150s text in
1994&ndash;95; it tries to give a Spivak-style rigorous presentation
in colorful mainstream-calculus-book format and reading level.
Horrible.  Take a look at it to see how badly written a mathematics
book can be.

<A NAME="e:bridges"><H2>Bridges to intermediate topics</H2></A>

<P>Springer-Verlag has just begun a new series of texts designed to
bring students gently into the realm of abstract mathematics.  While
there is no shortage of such books, these seem better than average
pedagogically; they are all quite talky, include complete solutions to
all exercises, and cover sensible (as opposed to traditional) sets of
topics.  The series is called SUMS, for Springer Undergraduate
Mathematics Series.  Two so far seem noteworthy:  Smith,
<i>Introduction to mathematics: algebra and analysis</i> and Johnson,
<i>Introduction to logic via numbers and sets</i>.  Give them a look.

<P><HR><A NAME="intermediate"><H1>INTERMEDIATE</H1></A>
Roughly, general rather than specialized texts in higher mathematics.
I would not hesitate to recommend any book here to honors
second-years, but they might not find easy going in some of them.

<H3>Contents</H3>
<UL>
  <LI><A HREF="#i:foundations">Foundations</A> (5)
  <LI><A HREF="#i:general-abstract-algebra">General abstract
  algebra</A> (7)
  <LI><A HREF="#i:linear-algebra">Linear algebra</A> (3)
  <LI><A HREF="#i:number-theory">Number theory</A> (5)
  <LI><A HREF="#i:combinatorics">Combinatorics and discrete
  mathematics</A> (1)
  <LI><A HREF="#i:real-analysis">Real analysis</A> (10)
  <LI><A HREF="#i:multivariable-calculus">Multivariable calculus</A> (2)
  <LI><A HREF="#i:complex-analysis">Complex analysis</A> (5)
  <LI><A HREF="#i:differential-equations">Differential equations</A> (2)
  <LI><A HREF="#i:point-set-topology">Point-set topology</A> (5)
  <LI><A HREF="#i:differential-geometry">Differential geometry</A> (4)
  <LI><A HREF="#i:classical-geometry">Classical geometry</A> (3)
</UL><HR>

<A NAME="i:foundations"><H2>Foundations</H2></A>

<H4>Halmos, <i>Naive set theory</i> <A href="http://www.ams.org/mathscinet-getitem?mr=56+%2311794">[MR 56 #11794]</A></H4>

<P>The best book for a first encounter with &ldquo;real&rdquo; set
theory.  Like everything Paul Halmos writes, it's stylistically
beautiful.  A very skinny book, broken into very short sections, each
dealing with a narrow topic and with an exercise or three.  It
requires just a little sophistication, but no great experience with
&ldquo;real&rdquo; math; we use this one for YSP kids sometimes too.

<H4>Fraenkel, <i>Abstract set theory</i> <A href="http://www.ams.org/mathscinet-getitem?mr=33+%235442">[MR 33 #5442]</A></H4>

<P>Fraenkel was the F in ZFC, and he gives a suitably rigorous
development of set theory from an axiomatic viewpoint.  Unfortunately,
for the philosophical foundations of the axioms he refers to another
book (Fraenkel and Bar-Hillel, <i>Foundations of set theory</i> <A href="http://www.ams.org/mathscinet-getitem?mr=49+%2310546">[MR 49 #10546]</A>),
which is missing from Eckhart Library.  Good for culture.

<H4>Ebbinghaus/Flum/Thomas, <i>Mathematical logic</i> <A href="http://www.ams.org/mathscinet-getitem?mr=95e%3A03002">[MR 95e:03002]</A></H4>

<P>The only logic book I can name off the top of my head, this is the
277 book.  I found it readable but boringly syntactic (well, maybe
that's elementary logic).

<h4>Enderton, <i>A mathematical introduction to logic</i> <A href="http://www.ams.org/mathscinet-getitem?mr=49+%232239">[MR 49 #2239]</A></h4>

<P>Look, another logic book!  This one might be preferable just
because there's much more talking about what's going on and less
unmotivated symbol-pushing than in E/F/T.  The flip side of that is,
the constructions may or may not be epsilon less precise.  I'm not a
logician; if you are, write some reviews so I can replace these lousy
ones!

<H4>Landau, <i>Foundations of analysis</i> <A href="http://www.ams.org/mathscinet-getitem?mr=12%2C397m">[MR 12,397m]</A></H4>

<P>This is the book that invented the infamous Landau
<i>&ldquo;Satz-Beweis&rdquo;</i> (theorem-proof) style.  There is
<i>nothing</i> in this book except the inexorable progression of
theorems and proofs, which is perhaps appropriate for a construction
of the real numbers from nothing, but makes horrible bathroom reading.
Read for culture.

<A NAME="i:general-abstract-algebra"><H2>General abstract
algebra</H2></A>

<P>The situation here is problematic, because there are many good
books which are just a little hard to swallo</i>w for an average 257
student, but precious few good ones below that.  But you learn by
doing, so here we go:

<H3>(Difficulty: moderate)</H3>

<H4>Dummit/Foote, <i>Abstract algebra</i> <A href="http://www.ams.org/mathscinet-getitem?mr=92k%3A00007">[MR 92k:00007]</A></H4>

<P><b>[PC]</b> I bought this for 257&mdash;I was at the age where I
uncritically bought all assigned texts (actually, I may still be at
that age; I don't recall passing on buying any course texts recently),
but as Chris knows the joke was on me, since we used the instructor's
lecture notes and not Dummit/Foote at all.  So I didn't really read it
that much at the time.  I have read it since, since it is one of two
general abstract algebra books in my collection.  I think it's an
excellent undergraduate reference in that it has something to say, and
often a lot to say, about precisely everything that an undergraduate
would ever run into in an algebra class&mdash;and I'm not even
exaggerating.  I would say this is a good book to have on your shelf
if you're an undergraduate because you can look up anything; I used it
this fall as a solid supplementary reference for character theory to
Alperin and Bell's <i>Groups and representations</i>, and it had an
amazing amount of material, all clearly explained.  [Warning: there is
an incorrect entry in one of the character tables; it's either A<sub>5</sub> or
S<sub>5</sub>, I can't remember which.]  Look elsewhere, particularly below, for
a good exposition of modules over a principal ideal domain; D/F's
exposition is convoluted and overly lengthy.  In fact, overall I would
use this book as a reference instead of a primary text, because the
idea of reading it through from start to finish scares me.  It also
has many, many good problems which develop even more topics (e.g.,
commutative algebra and algebraic geometry).

<H4>Herstein, <i>Topics in algebra</i> <A href="http://www.ams.org/mathscinet-getitem?mr=50+%239456">[MR 50 #9456]</A></H4>

<P>This is a classic text by one of the masters.  Herstein has
beautiful and elementary treatments of groups and linear algebra (in
the context of module theory).  But there is no field theory, and he
writes mappings on the right, which annoys many people.  Sometimes he
suffers from the same flaw of excessive elementarity as Spivak's
calculus book, but overall the treatment is quite pretty.  Many good
exercises.  (Not to be confused with <i>Abstract algebra</i> <A href="http://www.ams.org/mathscinet-getitem?mr=96j%3A00001">[MR 96j:00001]</A>, which is
a much-cut version for non-honors classes.)

<P><b>[PC]</b> But this is the book I would use if I were a
well-prepared undergraduate wanting to learn abstract algebra for the
first time.  Wonderful exposition&mdash;clean, chatty but not
longwinded, informal&mdash;and a very efficient coverage of just the
most important topics of undergraduate algebra. Think of it as a
slimmed down D/F.  &ldquo;No field theory&rdquo; is certainly an
exaggeration; the exposition there is quite brief, and the restriction
to fields of characteristic zero obscures the fact that much of the
theory presented, including the Galois theory, is the theory of
<i>separable</i> field extensions, but even so, this is still the book
I open first to remind myself about the Galois theory I'm supposed to
know.  The last main chapter of the book is quite lengthy and treats
linear algebra and canonical forms in detail, which is one of the
book's strongest features.  Also, there are many supplementary
topics&mdash;maybe Herstein really doesn't like field theory, since he
inserts a section on the transcendence of e early on in his field
theory chapter as something of a breather&mdash;but there's lots of
good stuff to warm the heart of someone who likes to see his algebra
applied to actual stuff, especially number-theoretic stuff; the famed
Two and Four Squares Theorems are both proved in here!

<H4>Artin, <i>Algebra</i> <A href="http://www.ams.org/mathscinet-getitem?mr=92g%3A00001">[MR 92g:00001]</a></H4>

<P>Artin's book is a nontraditional approach to undergraduate algebra,
emphasizing concrete computational examples heavily throughout.
Accordingly, linear algebra and matrix groups occupy the first part of
the book, and the traditional group-ring-field troika comes later.
This approach has the advantage of providing many nontrivial examples
of the general theories, but you may not want to wait that long to get
there.  Supposed to be well written, though I haven't read it
thoroughly.

<H3>(Difficulty: higher)</H3>

<H4>Jacobson, <i>Basic algebra</i> I <a href="http://www.ams.org/mathscinet-getitem?mr=86d%3A00001">[MR 86d:00001]</A></H4>

<P>Jacobson was my first real algebra book, and I retain an affection
for it.  The book is very densely written, and his prose has its own
beauty but is difficult to get much from at first.  The selection of
topics is interesting: chapters 1&ndash;4 cover groups, rings,
modules, fields (modules in the linear-algebra sense, that is, over
principal ideal domains), while chapters 5&ndash;8 cover extension
topics not usually found in general texts.  He deliberately avoids
modernist abstraction, preferring an explicit construction to a
universal property and a commutative diagram (although the universal
property is frequently given), and this complicates his notation and
prose at times, especially in the module chapter.  The field-theory
chapter is fantastic.  Some of the exercises are deliberately too
hard.

<H4>Hungerford, <i>Algebra</i> <a href="http://www.ams.org/mathscinet-getitem?mr=82a%3A00006">[MR 82a:00006]</a></H4>

<P>Many people like this book, but I don't.  Hungerford covers the
standard topics from group, ring, module, and field theory, with a
little additional commutative ring theory and the Wedderburn theory of
algebras.  The field-theory chapter is horrible, and the rest of the
book is okay but doesn't excite me.  (And the typesetting is bad.)

<H4>Lang, <i>Algebra</i> <a href="http://www.ams.org/mathscinet-getitem?mr=86j%3A00003">[MR 86j:00003]</a></H4>

<P>Well, do you like Serge Lang books, or not?  Like every other Serge
Lang book, this one is uncompromisingly modern, wonderfully
comprehensive, and unpleasantly dry and tedious to read.  Unlike most
other Serge Lang books, this one has exercises, at least.

<H4>Mac Lane/Birkhoff, <i>Algebra</i> <a href="http://www.ams.org/mathscinet-getitem?mr=89i%3A00009">[MR 89i:00009]</a></H4>

<P>I keep recommending this book to people because it's the only hard
one whose contents correspond well to the 257-8-9 syllabus, and also
because I like Mac Lane's treatment of linear and multilinear algebra.
Mac Lane and Lang are the only books in this group which treat
multilinear (tensor) algebra at all, and believe me, you'll need it
eventually.  Worth a look to see whether you find Mac Lane's style
congenial.  Not to be confused with Birkhoff/Mac Lane, <i>A survey of
modern algebra</i> <a href="http://www.ams.org/mathscinet-getitem?mr=31+%232250">[MR 31 #2250]</a> (a much shorter and easier book).

<P><b>[BR]</b> I used Mac Lane/Birkhoff's book pretty heavily in Math
257 and 258.  Unlike most algebra books I've seen, they don't put all
the group theory at the beginning and all of the field theory at the
end, but prefer to develop each topic a little bit at a time and then
develop it with more depth later.  As a result, this book is hard to
use as a reference.  You can't get past rings without tackling
categories and universal constructions which are used heavily
throughout the remainder of the text.  However, their treatment of
categorical algebra is one of the more readable introductions to the
theory I've come across.

<A NAME="i:linear-algebra"><H2>Linear algebra</H2></A>

<H4>Halmos, <i>Finite dimensional vector spaces</i> <a href="http://www.ams.org/mathscinet-getitem?mr=53+%2313258">[MR 53 #13258]</a></H4>

<P>This is a linear algebra book written by a functional analyst, and
the crux of the book is a treatment of the spectral theorem for
self-adjoint operators in the finite-dimensional case.  It's a
beautiful, wonderful book, but not a very good reference for
traditional linear algebra topics or applications.  You also have to
read a fair distance before you even see a linear map, and the
exercises are mostly too easy, with a few too hard.  But this book was
where I first learned about tensor products, and why the matrix
elements go the way they do and not the other way (Halmos is very
careful on this point).

<P><b>[PC]</b> I own this book and read through it often, but it's
never taught me linear algebra per se.  Let's agree that it's too
abstract for a reasonable first introduction to linear algebra; it's
really meant for students who already know (some) linear algebra to
read through and appreciate one particular, and particularly elegant,
presentation of the material.  If you want to know about the linear
algebra which surrounds functional analysis, then by all means read
this book, but much of the material is nonstandard and a bit curious
from the perspective of mainstream linear algebra; projections seem to
be the most important linear map, and there are many sections lovingly
devoted to commuting projections, decomposing projections, etc.  I
still am not sure why Halmos deifies the [,] as much as he does, and
quite honestly, I would learn multilinear algebra anywhere but here.

<H4>Curtis, <i>Abstract linear algebra</i> <A href="http://www.ams.org/mathscinet-getitem?mr=91j%3A15003">[MR 91j:15003]</a></H4>

<P>If you can stand terrible typesetting and an unexciting prose
style, this tiny little book is a good rigorous reference for
traditional linear algebra (i.e., it doesn't assume you're a tree).  A
nice bonus at the end is the Wedderburn theorem for division algebras
over <b>R</b>, although the lack of sophistication makes for some
unmotivated technical carpentry.  I look in here whenever I can't
remember what a positive-definite matrix is.

<H4>Greub, <i>Linear algebra</i> <a href="http://www.ams.org/mathscinet-getitem?mr=51+%235615">[MR 51 #5615]</a> and <i>Multilinear algebra</i> <a href="http://www.ams.org/mathscinet-getitem?mr=80c%3A15017">[MR 80c:15017]</a></H4>

<P>You may never need The Book on linear algebra.  But one day, you
may just have to know fifteen different ways to decompose a linear map
into parts with different nice properties.  On that day, your choices
are Greub and Bourbaki.  Greub is easier to carry.  End of story.

<A NAME="i:number-theory"><H2>Number theory</H2></A>

<H4>Ireland/Rosen, <i>A classical introduction to modern number
theory</i> <a href="http://www.ams.org/mathscinet-getitem?mr=92e%3A11001">[MR 92e:11001]</a></H4>

<P>The first half is a coherent, systematic development of elementary
number theory, assuming the basics of algebra.  In the second half the
authors explore more advanced topics of an algebraic/geometric flavor
(zeta functions, L-functions, algebraic number fields, elliptic
curves).  Lots of exercises.  This book helped make number theory make
sense to me.  You will find many introductory number theory texts
pitched below I/R, but if you can read I/R, ignore the easy ones.

<P><b>[PC]</b> Yes, this is the standard and to my knowledge the best
number theory text that is modern, broad, and reasonably elementary.
It's a strange book in that it's really not written at any one
level&mdash;if you've heard of something called unique factorization,
you'll find the first few chapters easygoing material, but the
algebraic sophistication rises slowly but surely throughout the book.
Eventually you need to be comfortable with rings, fields and Galois
theory at the undergraduate level, but they tell you at the beginning
of the chapter when they require more background than before.  There's
an awful lot in here; this was my course text for Math 242 and I used
it as one of the texts in a reading class on number theory, and I
still haven't read through all the chapters.  It's a great example of
a book in which the authors have tried and succeeded in bringing
advanced material down to the undergraduate level.  Some good
historical notes, as any self-respecting number theory text should
contain.  Recommended highly.

<H4>Burn, <i>A pathway into number theory</i> <a href="http://www.ams.org/mathscinet-getitem?mr=97g%3A11001">[MR 97g:11001]</a></H4>

<P><b>[BB]</b> The book is composed entirely of exercises leading the
reader through all the elementary theorems of number theory.  Can be
tedious (you get to verify, say, Fermat's little theorem for maybe 5
different sets of numbers) but a good way to really work through the
beginnings of the subject on one's own.

<H4>Hardy/Wright, <i>Introduction to number theory</i> <a href="http://www.ams.org/mathscinet-getitem?mr=81i%3A10002">[MR 81i:10002]</a></H4>

<P>This is the classic, and Hardy is one of the great expository
writers of mathematics.  However, I remember that the last time I
looked at this book it made no sense to me.  If you like number theory
you should probably at least look at it.

<P><b>[PC]</b> Oh, here I must fervently disagree (well, okay, maybe
it didn't make sense to you at the time, but please go ahead and look
again).  I say that <i>any</i> student of mathematics should have this
book on their shelf.  Here's H/W's game: they explain number theory to
people who can follow mathematical proofs but have no prior exposure
to the subject or <i>any</i> advanced machinery whatsoever&mdash;hmm,
maybe a little calculus at times, but not always.  The one thing they
do use is a little asymptotic growth notation, i.e., O, o, and the
squiggly line, and for some reason they assume that people will know
all about this without much comment.  I seem to recall that one
chapter towards the beginning is confusing because of this, and when I
first bought the book it stymied me (I was sixteen at the time).  But
it's written so that you don't have to read it in order: they develop
just enough theory about almost every branch of (elementary) number
theory so that you can see interesting theorems proved.  I have jumped
around a lot, but over the years I think I've read almost every
chapter.  I really think it's the #1 &ldquo;cultural enrichment&rdquo;
book for math students.

<H4>Chandrasekharan, <i>Introduction to analytic number theory</i> <a href="http://www.ams.org/mathscinet-getitem?mr=40+%232593">[MR 40 #2593]</a></H4>

<P><b>[PC]</b> Recommended to me by none other than Professor
Narasimhan himself, it's actually a very elementary and readable
introduction to the classic theorems of analytic number theory:
Chebyshev's Theorem, Bertrand's Postulate, uniform distribution,
Dirichlet's Theorem and the Prime Number Theorem.  Requires
epsilonics and just a little bit of complex function theory.

<H4>Apostol, <i>Introduction to analytic number theory</i> <a href="http://www.ams.org/mathscinet-getitem?mr=55+237892">[MR 55 #7892]</a></H4>

<P><b>[PC]</b> If you've been reading this list, you know from Chris
that Apostol writes terribly dry books.  I've never read anything by
him but this one, and it's fine, a bit more elementary than
Chandrasekharan and easier to get your hands on (Apostol is a UTM;
Chandrasekharan is an out of print Springer international edition).
It starts out with a nice introduction to arithmetic functions,
including the convolution product, and it covers much the same as the
above, only a bit less briskly.  A quick route to the proofs of the
greatest theorems of 19th century mathematics.

<A NAME="i:combinatorics"><H2>Combinatorics and discrete
mathematics</H2></A>

<H4>Graham/Knuth/Patashnik, <i>Concrete mathematics</i> <a href="http://www.ams.org/mathscinet-getitem?mr=97d%3A68003">[MR 97d:68003]</a></H4>

<P>The first chapter of Knuth's immortal work <i>The art of computer
programming</i> is an extensive study of combinatorics and
asymptotics.  G/K/P is an expanded and friendlier version, which
emphasizes teaching the reader to solve things, rather than just
showing how they are done.  Contains many funny marginal notes from
students in the Stanford class which gave birth to the book, as well
as tons of great exercises.  Not a reference work.

<A NAME="i:real-analysis"><H2>Real analysis</H2></A>

<H3>(Elementary level: metric spaces, continuity, differentiation)</H3>

<H4>Rudin, <i>Principles of mathematical analysis</i> <a href="http://www.ams.org/mathscinet-getitem?mr=52+%235893">[MR 52 #5893]</a></H4>

<P>The first eight chapters of this little book form the best,
cleanest exposition of elementary real analysis I know of, although
few UC readers will have much use for the chapter on Riemann-Stieltjes
integration.  Like Rudin's other books, it is broken into bite-size
pieces, so you can prove every statement in the book on your own if
you're self-studying.  If that isn't enough, there is a large
collection of challenging exercises.  Some people think Rudin is too
skinny and streamlined, but I think it's beautiful.  (Ignore chapters
9 and 10, which are a confusing and insufficiently motivated
development of multivariable calculus.  Chapter 11 is all right for
Lebesgue integration, but there are better treatments elsewhere.)

<P><b>[PC]</b> I agree 100% with what Chris says, but I want to add my
voice that this is (through chapter 8) the cleanest exposition I have
ever seen.  I still flip back to this to check things out.

<P><b>[BR]</b> I must insist that Chapters 9 and 10 are not THAT bad.
They're worth revisiting if you are tired of Spivak and do Carmo.

<H4>Apostol, <i>Mathematical analysis</i> <a href="http://www.ams.org/mathscinet-getitem?mr=49+%239123">[MR 49 #9123]</a></H4>

<P>Covers the same material as Rudin, plus a little complex analysis.
Apostol assumes (hence, engenders) less maturity on the reader's part,
writing most arguments out in &ldquo;advanced calculus&rdquo; detail
rather than &ldquo;real analysis&rdquo; detail, if that makes sense.
I find it terribly dry.  Nevertheless the book is careful and
comprehensive, with many exercises.

<H4>Gelbaum/Olmsted, <i>Counterexamples in analysis</i> <a href="http://www.ams.org/mathscinet-getitem?mr=30+%23204">[MR 30 #204]</a></H4>

This little book contains a long list of examples, of strange objects
which contradict the things that you think should be true but aren't.
It starts off at a very elementary level and gradually builds up to
include the Lebesgue theory and <b>R</b><sup>n</sup>.  A good thing to have
around on your first or second trip through analysis.  (See also the
authors' <i>Theorems and counterexamples in mathematics</i> <a
href="http://www.ams.org/mathscinet-getitem?mr=92b%3A00003">[MR
92b:00003]</a>, an expanded version covering fields other than
analysis.)

<H3>(Intermediate level: normed spaces, Lebesgue integration)</H3>

<H4>Kolmogorov/Fomin, <i>Introductory real analysis</i> <a href="http://www.ams.org/mathscinet-getitem?mr=51+%2313617">[MR 51 #13617]</a></H4>

<P>When I started 207 I couldn't see why the material of this book was
analysis: here was set theory, some linear algebra, some stuff about
normed linear spaces, a little functional analysis... oh, here's that
cool integral everyone talks about, but where are the derivatives?
Now I know why it's analysis, of course, but the book as a whole is
still a perplexing beast to the inexperienced.  I think the primary
reason it remains a text for 207 is that it costs $13, so why not?
The style is distinctively Russian, which puts me off but turns other
people on.  Extended applications appear occasionally to lend context,
but on the whole there is little motivation (and few exercises).  The
book is also difficult to use as a reference work, because the authors
develop only the results they need to get where they're going.

<P><b>[PC]</b> Agreed.  But it's cheap and though you may wonder why
you're learning so much functional analysis before you see a Lebesgue
integral, it's still clear and easy to read, so there's no reason why
you shouldn't own it.

<H4>Haaser/Sullivan, <i>Real analysis</i> <a href="http://www.ams.org/mathscinet-getitem?mr=92a%3A00006">[MR 92a:00006]</a></H4>

<P>Covers the same material as K/F, with the addition of a chapter
relating differentiation to Lebesgue integration (the fundamental
theorems of calculus).  H/S use the Daniell integral rather than K/F's
concrete, bare-hands construction of Lebesgue measure; it's probably
good to do it by hand once, but after that forget it.  The sequence of
topics makes a little more sense than K/F, although the chapter on
inner product spaces is lonely at the end, where it lives because they
want to do Fourier series.  But the book is written in a ho-hum
style, and the exercises are too easy.  In this H/S shares the flaw of
many books at this level, of making too big a deal of a little bit of
abstraction which might be new to the reader.  I went straight from
little Rudin to big Rudin without much of a stop for either of these
books.

<H4>Hewitt/Stromberg, <i>Real and abstract analysis</i> <a href="http://www.ams.org/mathscinet-getitem?mr=51+%233363">[MR 51 #3363]</a></H4>

<P>This is an old, classic book which is worth a look.  They develop
many concrete classical topics (all those things like Legendre
polynomials that you were always curious about) as exercises.

<H4>Dieudonn&eacute;, <i>Foundations of modern analysis</i> <a href="http://www.ams.org/mathscinet-getitem?mr=50+%231782">[MR 50 #1782]</a></H4>

<P>This book is a strange bird, the first volume of a nine(!)-volume
treatise by one of the original Bourbakistes.  I can't really describe
it except to say that it's very formalistic, it has many good
exercises, it's very hard to relate to other treatments of the
subject, and it made a big impression on me.

<H3>(Graduate level: measure theory, basic functional analysis)</H3>

<H4>Rudin, <i>Real and complex analysis</i> <a href="http://www.ams.org/mathscinet-getitem?mr=88k%3A00002">[MR 88k:00002]</a></H4>

<P>The first half is the standard reference for real analysis (the
second half is reviewed below).  It's a very clean treatment of the
topics it covers, again in bite-size pieces and with many
<i>challenging</i> exercises.  Sometimes I get frustrated with the
lack of motivation, or with Rudin's habit of proving exactly the
lemmas he needs to do something, without any context for the results.
Nevertheless it's a good reference or self-study book.  Topics:
Integration and L<sup>p</sup> spaces, Banach and Hilbert spaces, Radon-Nikodym
theorem and differentiation, Fubini's theorem, Fourier transforms.

<P><b>[PC]</b> Yes, how wonderful that there's one book whose first
half contains all the analysis that you'll ever need to know!  This
book is advanced and the exposition is austere (&ldquo;which gives
(5).  Applying (3) to (4), we get (6)&rdquo;) but it is absolutely
crystalline in its clarity (exception: is its proof of the L<sup>2</sup>
inversion theorem for Fourier transforms valid?  I'm not so sure.)
Isn't this the one math book that every student must buy sooner or
later (aside from Hardy and Wright, of course)?  Some rainy day you'll
discover that the book has a second half and find some very
interesting theorems in there, but don't confuse it with a course on
complex analysis, because it's a weird-ass treatment of complex
analysis viewed through the eyes of a conventional analyst.  Think of
it as a bonus.

<H4>Lang, <i>Real and functional analysis</i> <a href="http://www.ams.org/mathscinet-getitem?mr=94b%3A00005">[MR 94b:00005]</a></H4>

<P>Another Serge Lang book, but a Serge Lang book is about the only
place you'll find the inverse function theorem systematically treated
for Banach spaces (except Dieudonn&eacute;, and Lang was a Bourbakiste
too).

<H4>Royden, <i>Real analysis</i> <a href="http://www.ams.org/mathscinet-getitem?mr=90g%3A00004">[MR 90g:00004]</a></H4>

<P>Royden is like Hungerford for me: a lot of people like it, but it
annoys me for a number of semi-silly reasons.  He denotes the empty
set by 0 (zero) and the zero element of a vector space by lowercase
theta.  He proves many theorems three times in gradually increasing
generality.  He leaves whole proofs to the exercises, and then depends
on them later in the text.  And I don't like his construction of
Lebesgue integration.  (Nyaah, so there.)

<P><b>[BR]</b> This is such a terrible book!  He leaves the hardest
theorems to the reader and proves some really simple-minded things
with too much machinery.  For example, he assigns the Urysohn lemma
for normal spaces as an exercise for the reader and then has to use
the Baire category theorem to show that on Banach spaces, linear
operators are continuous iff they commute with taking limits.  If you
have to take 208 or 272, find a supplementary text.  You'll be happy
you did.

<A NAME="i:multivariable-calculus"><H2>Multivariable calculus</H2></A>

<H4>Spivak, <i>Calculus on manifolds</i> <a href="http://www.ams.org/mathscinet-getitem?mr=35+%23309">[MR 35 #309]</a></H4>

<P>This is the book everybody gets in differentiation and integration
in <b>R</b><sup>n</sup>, and it's a pretty good one, although the integration
chapters are hard to read&mdash;maybe it was just my first encounter with
exterior algebra that made it hard.  As usual for Spivak books, clear
exposition and lots of nice exercises.  Unfortunately this one is old
enough to be annoyingly typeset.

<P><b>[PC]</b> I don't really like this book, and I'm a big fan of
Spivak in general.  Does anybody else think that this rigorous
multivariable Riemann integral theory is a dinosaur?  And when Spivak
starts talking about chains (in chapter four, I think), I don't know
what the hell he's talking about.  Presumably you could ignore that
chapter and use the book as an introduction to differential forms.  I
can't suggest a substitute at the moment, other than Spivak's
<i>Comprehensive introduction</i> volume 1, which is a wonderful book
but which I still wouldn't want to read as a first introduction to
forms.  Come to think of it, I love forms to death, but maybe they're
just plain confusing the first time around...

<H4>do Carmo, <i>Differential forms and applications</i> <a href="http://www.ams.org/mathscinet-getitem?mr=95j%3A58002">[MR 95j:58002]</a></H4>

<P>This skinny yellow book has replaced Munkres's <i>Analysis on
manifolds</i> <a href="http://www.ams.org/mathscinet-getitem?mr=92d%3A58001">[MR 92d:58001]</a> as the text for 274, and I'm not sure it's an
improvement.  It's more like a modernized <i>Calculus on
manifolds</i>.  I haven't done more than glance through it, but the
notation is reputedly horrible, and Spivak is definitely a superior
expositor.

<A NAME="i:complex-analysis"><H2>Complex analysis</H2></A>

<H4>Ahlfors, <i>Complex analysis</i> <a href="http://www.ams.org/mathscinet-getitem?mr=80c%3A30001">[MR 80c:30001]</a></H4>

<P>Ahlfors has been the standard text for complex function theory for
quite some time.  I like it, but he's very classical and concrete in
outlook: nary a function space or a norm in the whole book.  The
exposition is a classic, though.

<P><b>[PC]</b> Everyone lists it; do people actually read it?  I'd use
Conway instead.

<H4>Conway, <i>Functions of one complex variable</i> I <a href="http://www.ams.org/mathscinet-getitem?mr=80c%3A30003">[MR 80c:30003]</a></H4>

<P>This book starts very, very slow and easy, so if you're rusty on
metric spaces or real-variable theory you have no need to worry.
Conway's style is to prove things very thoroughly, but relegate the
occasional proof to the exercises.  The text is more modern than
Ahlfors; Conway proves Runge's theorem using Banach space techniques
(well, he's an operator theorist).  I like the book more for this
reason, but I finally sold my copy because the slow pace got to me.

<P><b>[PC]</b> I like the book, but I hear your criticisms.  The
chapter on convergence in the compact-open topology, arguably the most
important topic in the whole book, is marred by the fact that he mixes
metric space theory which is perfectly general with the theory of
complex functions.  His chapter on Riemann surfaces sort of annoys me
too, for the same reason.  Maybe just a bit of reorganization would
improve this book.  But he covers all the theorems that an
undergraduate needs to know (and a little more), and he does it
without using fancy machinery of any sort: no fundamental groups, no
differential forms, no deep theorems from real analysis.  [CJ: The
Hahn-Banach theorem isn't a deep theorem from real analysis?]  Still,
I can't help but think that the great American complex analysis book
has yet to be written.

<H4>Narasimhan, <i>Complex analysis in one variable</i> <a href="http://www.ams.org/mathscinet-getitem?mr=2002e%3A30001">[MR 2002e:30001]</a></H4>

<P>As we might expect from the famed freshman-eating Narasimhan, this
book is much quicker-paced and covers more topics than either of the
two above (including a chapter on several variables).  Sadly, there
are no exercises, but the book is a good reference work. (Note: Exercises
were added to the second edition by the junior author Yves Nievergelt.)

<H4>Rudin, <i>Real and complex analysis</i> <a href="http://www.ams.org/mathscinet-getitem?mr=88k%3A00002">[MR 88k:00002]</a></H4>

<P>Rudin's second half is a treatment of complex analysis even more
modern than Conway, but resolutely non-geometric.  I never really got
along with it, for the second reason; also, the selection of topics
after the canonical material feels a little random.  (Rudin's aim was
to bring out the unifying threads in real and complex analysis; thus
there is a chapter on Banach algebras near the end.)  However, the
style is still crystalline, and the exercises are still excellent.
Best for confirmed analysts.

<H4>Palka, <i>An introduction to complex function theory</i> <a href="http://www.ams.org/mathscinet-getitem?mr=92b%3A30001">[MR 92b:30001]</a></H4>

<P><b>[YU]</b> The author follows Ahlfors's approach and thus the book
is very geometric.  After reading this book, I began to like complex
function theory.  It contains lots of interesting exercises as well as
routine ones.

<A NAME="i:differential-equations"><H2>Differential equations</H2></A>

<H4>Arnold, <i>Ordinary differential equations</i> <a href="http://www.ams.org/mathscinet-getitem?mr=93b%3A34001">[MR 93b:34001]</a></H4>

<P>Yes, Virginia, there is an interesting geometric theory of
differential equations (of course!), not just the stuff you see in
those engineering texts: stuff about stable and unstable points or
manifolds, and other things with a dynamical-systems flavor.
Nevertheless there is substantial material on how to reduce a
differential equation to linear form and solve it, although no Laplace
transform techniques or the like.  Arnold explains it all coherently
at an advanced-calculus level (manifolds appear at the end), complete
with many beautiful diagrams.  Another distinctively Russian
book&mdash;read all the ones I describe that way, and you'll see what
I mean.  The third edition is substantially different from the second
(which I have): the manifolds material is much expanded, and the
typesetting is not so nice.

<H4>Hurewicz, <i>Lectures on ordinary differential equations</i> <a href="http://www.ams.org/mathscinet-getitem?mr=91k%3A34001">[MR 91k:34001]</a></H4>

<P>A tiny book which covers material similar to Arnold, but more
concisely.  I haven't read it but it's frequently referenced, and
worth a look if you need to know the basic theorems.  (If all you need
is the basic existence-uniqueness theorem for ODEs, it's also in
Spivak volume 1 or Lang, <i>Real and functional analysis</i>.)

<A NAME="i:point-set-topology"><H2>Point-set topology</H2></A>

<H4>Munkres, <i>Topology: a first course</i> <a href="http://www.ams.org/mathscinet-getitem?mr=57+%234063">[MR 57 #4063]</a></H4>

<P>Munkres's book is a wonderful first encounter with topology; in
fact it begins slowly enough to be a first encounter with abstract
mathematics (after a traditional advanced calculus course).  Every
abstraction is carefully motivated, and there are tons of examples,
pictures, and exercises.  This is one of those books you could hand to
a bright student of any age who knew some calculus (not a bad book to
choose if you're coming back to mathematics at age 35).  Most of the
book is the traditional analysis-topology material, but there is a
long last chapter on the fundamental group which covers enough to
prove the Jordan curve theorem. (Note: in the second edition, not yet
reviewed by MR, the material on algebraic topology has been
substantially expanded.)

<P><b>[PC]</b> Yes, Munkres deserves to be the standard undergraduate
point-set book.  It doesn't have everything, but it has most of the
standard topics and it's relentlessly clear.

<H4>Willard, <i>General topology</i> <a href="http://www.ams.org/mathscinet-getitem?mr=41+%239173">[MR 41 #9173]</a></H4>

<P>But Willard is my topology book of choice.  The level of
abstraction is deliberately higher, and the book is better organized
as a reference than Munkres.  It's not nearly as friendly, but it's
still clear and well-written (I think an unclear point-set topology
book is probably no longer a point-set topology book).  Willard is
probably the best modern reference for analysis-topology, where
&ldquo;modern&rdquo; means &ldquo;excluding Kelley&rdquo; (see below).
You can learn from it too; it's organized bite-size like a Rudin book,
so you can prove all but the hard theorems on your own (I did this
with an initial segment, and learned a lot).

<H4>Kelley, <i>General topology</i> <a href="http://www.ams.org/mathscinet-getitem?mr=51+%236681">[MR 51 #6681]</a></H4>

<P><b>[PS]</b> Let me just say that Kelley's book on topology is
horribly old-fashioned&mdash;I know because my advisor is forcing me
to read it.  Half the topics are things which I don't think are as
important as they used to be.  Nets, filters?  I guess they're
interesting in and of themselves.  On the upside, it does have a nice
appendix covering the rudiments of set theory.

<P><b>[CJ]</b> It is old-fashioned, but it's still the best book on
topology for functional analysis, bar none.  Nets are surprisingly
necessary in infinite-dimensional topological vector spaces!  The
occasional proof is easier to read once recast in modern language, but
doing so is a good learning exercise anyway.  And Kelley has the nice
habit (emulated less successfully by Willard) of treating substantial
pieces of analysis as exercises; two of the exercises to Chapter 2 are
titled &ldquo;Integration theory, junior grade&rdquo; and
&ldquo;Integration theory, utility grade&rdquo;.  It's really an
analysis book disguised as a point-set topology book, but then much of
functional analysis is really general topology on spaces that happen
to be vector spaces too.

<H4>Steen/Seebach, <i>Counterexamples in topology</i> <a href="http://www.ams.org/mathscinet-getitem?mr=96k%3A54001">[MR 96k:54001]</a></H4>

<P>This is a topology &lsquo;anticourse&rsquo;: a collection of all
the screwed-up topological spaces which provide limiting
counterexamples to all those point-set topology theorems with
complicated hypotheses.  It's a classic just for the content, but
pretty well written too.  This book and Gelbaum/Olmsted (above) are
two parts of what should someday be the big book of counterexamples to
everything.  Read it and see just what you avoid by sticking to
differentiable manifolds.

<P><b>[BR]</b> Steen and Seebach have catalogued 143 of the most
disgusting pathological topological creatures.  They are invaluable
for when you're first learning point set topology and need to
understand why the definitions are necessary.  They can also come in
handy on tests: I used the one-point compactification of an
uncountable discrete space three times on my Math 262 final.  The text
used for 262, Munkres, relies on three counterexamples to disprove
everything: the Sorgenfrey line, S<sub>&Omega;</sub> and I x I in the dictionary
order. Steen and Seebach let you know that there are tons of other
beastly topological spaces which violate the laws of common sense.

<H4>Dugundji, <i>Topology</i> <a href="http://www.ams.org/mathscinet-getitem?mr=57+%2317581">[MR 57 #17581]</a></H4>

<P><b>[YU]</b>This is a point-set topology book.  Less elementary than
Munkres, but useful as a reference book for grad students.

<A NAME="i:differential-geometry"><H2>Differential geometry</H2></A>

<H4>Guillemin/Pollack, <i>Differential topology</i> <a href="http://www.ams.org/mathscinet-getitem?mr=50+%231276">[MR 50 #1276]</a></H4>

<P>I didn't understand transversality at all until I saw this book.
It's a very geometric (as opposed to formalistic), down-to-earth
introduction to some of the most mystical areas of smooth manifold
theory: transversality and intersection theory.  Abstraction is
avoided (manifolds are defined as embedded in Euclidean space, which
annoys me just a bit), but without hand-waving important distinctions
(they are careful to point out that for noncompact manifolds, an
injective immersion need not be an embedding, that is, proper too).
The last chapter treats integration and Stokes's theorem, but that's
not what anyone reads the book for.  Beautifully written, and fills an
important hole in Spivak volume 1.

<H4>do Carmo, <i>Differential geometry of curves and surfaces</i> <a href="http://www.ams.org/mathscinet-getitem?mr=52+%2315253">[MR 52 #15253]</a></H4>

<P>We used this book for Corlette's differential geometry seminar two
years ago (293).  I didn't like it all that much because do Carmo is
careful to keep the book to a post-advanced-calculus level: everything
takes place in <b>R</b><sup>3</sup>, no vector bundles, lots of componentwise
calculations.  Nevertheless it's a nice treatment of the classical
theory of curves and surfaces in space.  Read it if you want to know
about the Gauss map or the two fundamental forms, but don't want to
work all the way through Spivak volume 2.

<H4>Spivak, <i>A comprehensive introduction to differential
geometry</i>, 1 <a href="http://www.ams.org/mathscinet-getitem?mr=82g%3A53003a">[MR 82g:53003a]</a></H4>

<P><b>[PC]</b> Volume 1 is the best introduction to smooth manifold
theory and differential topology that I know of.  Every chapter of
this book has come in handy for me at one time or another.  Ben and I
like to describe the book as &ldquo;locally readable&rdquo;: his
exposition is very careful, but sometimes he takes too damn long to
explain a single concept.  Luckily, despite Spivak's efforts to the
contrary, you can flip around and read chapter by chapter, and I
recommend this.  There is so much good stuff in here.

<P><b>[CJ]</b> Buy it and read it over and over and over.  Don't skip
the exercises because that's where he puts all the freaky examples.
It's true that sometimes he talks too much, but for the loving detail
in which he lays out difficult concepts, he can be forgiven.

<H4>Spivak, <i>A comprehensive introduction to differential
geometry</i>, 2 <a href="http://www.ams.org/mathscinet-getitem?mr=82g%3A53003b">[MR 82g:53003b]</a></H4>

<P>As Spivak puts it at the beginning, &ldquo;Volume 1 dealt with the
&lsquo;differential&rsquo; part; in this volume we finally get down to
some geometry.&rdquo; Volume 2 treats the classical theory of curves and
surfaces using the modern machinery developed in the first volume,
which makes it (for me) a more comfortable read than do Carmo.  Spivak
is careful to motivate everything historically; surface theory is
introduced by a long walk through Gauss's <i>General investigations of
curved surfaces</i> (you should really have a copy of it to read this
book), and the second half of the book goes through the (convoluted)
stages of evolution of the definition of a connection.  Not easy
reading but every bit as rewarding as Volume 1.  Unfortunately there
are almost none of the wonderful exercises which characterize the
first volume.

<A NAME="i:classical-geometry"><H2>Classical geometry</H2></A>

<H4>Coxeter, <i>Introduction to geometry</i> <a href="http://www.ams.org/mathscinet-getitem?mr=90a%3A51001">[MR 90a:51001]</a></H4>

<P>This is an interesting book which I can't really describe.  It
contains a number of short treatments of undeniably geometric but
nontraditional topics; one fascinating application is the relation
between phyllotaxis (the arrangement of plants' leaves around the
stem) and generalized Fibonacci-type numbers.  Read for culture.

<H4>Hilbert, <i>Foundations of geometry</i> <a href="http://www.ams.org/mathscinet-getitem?mr=43+%231019">[MR 43 #1019]</a></H4>

Hilbert was very interested in finding coherent, minimal axiom systems
for parts of mathematics; he was probably inspired by the long debate
over Euclid's parallel postulate and the discovery in the late 19th
century of consistent non-Euclidean geometries.  (The G&ouml;del
incompleteness theorems solved negatively one of Hilbert's famous
problems.)  In this book Hilbert described a correct and complete
axiom system for Euclidean geometry, with the dependence relations
between axioms exhaustively determined, and then carefully derived
most of Euclid from it.  It's not a particularly fun read but its
existence is philosophically interesting.

<H4>Hartshorne, <i>Geometry: Euclid and beyond</i> <a href="http://www.ams.org/mathscinet-getitem?mr=2001h%3A51001">[MR 2001h:51001]</a></H4>

<P>The algebraic geometer of the famed book from hell (see below)
recently finished another modern-Euclid book.  It's a sophisticated
modern explication for a guided reading of much of Euclid's <i>Elements</i>.
Very neat.

<P><HR><A NAME="advanced"><H1>ADVANCED</H1></A>
Specialized works, difficulty level unbounded above.

<H3>Contents</H3>
<UL>
  <LI><A HREF="#a:foundations">Foundations</A> (1)
  <LI><A HREF="#a:problem-solving">Problem solving</A> (1)
  <LI><A HREF="#a:general-abstract-algebra">General abstract
  algebra</A> (1)
  <LI><A HREF="#a:group-theory">Group theory and representations</A> (5)
  <LI><A HREF="#a:ring-theory">Ring theory</A> (4)
  <LI><A HREF="#a:commutative-homological">Commutative and homological
  algebra</A> (5)
  <LI><A HREF="#a:number-theory">Number theory</A> (5)
  <LI><A HREF="#a:combinatorics">Combinatorics and discrete
  mathematics</A> (3)
  <LI><A HREF="#a:measure-theory">Measure theory</A> (2)
  <LI><A HREF="#a:probability">Probability</A> (1)
  <LI><A HREF="#a:functional-analysis">Functional analysis</A> (5)
  <LI><A HREF="#a:complex-analysis">Complex analysis</A> (6)
  <LI><A HREF="#a:harmonic-analysis">Harmonic analysis</A> (5)
  <LI><A HREF="#a:differential-equations">Differential equations</A> (4)
  <LI><A HREF="#a:differential-topology">Differential topology</A> (3)
  <LI><A HREF="#a:algebraic-topology">Algebraic topology</A> (7)
  <LI><A HREF="#a:differential-geometry">Differential geometry</A> (6)
  <LI><A HREF="#a:geometric-measure-theory">Geometric measure
  theory</A> (4)
  <LI><A HREF="#a:algebraic-geometry">Algebraic geometry</A> (5)
</UL><HR>

<A NAME="a:foundations"><H2>Foundations</H2></A>

<H4>Mac Lane, <i>Categories for the working mathematician</i></H4>

<P>Pete Clark isn't convinced that the working mathematician needs any
category theory at all, but I definitely am!  Of course it depends on
whether you're interested in something heavily homological, but most
people will need at least the basics of adjoints and limits sometime.
The book covers substantially more than that, but because examples are
drawn from some advanced stuff (rings and Lie algebras appear in the
first chapter) you need a fair amount of background to read it.
Noteworthy is a section near the end entitled &ldquo;All concepts are
Kan extensions&rdquo;.  Most books on homological algebra will contain
a brief summary of category theory, as does Jacobson's <i>Basic
algebra</i> II; here you can find it laid out in more detail.

<A NAME="a:problem-solving"><H2>Problem solving</H2></A>

<H4>P&oacute;lya/Szeg&ouml;, <i>Problems and theorems in analysis</i> I and II</H4>

<P>These are very old books of very good problems, mostly from
analysis, with complete solutions.  They're old-fashioned of course,
but the polite word is &ldquo;classical&rdquo;; worth reading for
culture, to prepare for your quals, or (important!) to see if you can
still do concrete calculations after four years of brainwashing by
abstraction.  (Anyone want to compute the n-Hausdorff measure of S<sup>n</sup>
in <b>R</b><sup>n+1</sup>?)

<A NAME="a:general-abstract-algebra"><H2>General abstract
algebra</H2></A>

<H4>Jacobson, <i>Basic algebra</i> II</H4>

<P>This is perhaps the only really advanced general-algebra book; it
contains chapters on categories, universal algebra, modules and module
categories, classical ring theory, representations of finite groups,
homological algebra, commutative algebra, advanced field theory...
Readability is uniformly low (unless you really like Jacobson's prose
style) and the quality (&ldquo;sanity&rdquo;) of the treatments
varies; I'd look anywhere else for group representation theory, but as
Jacobson is a ring theorist, the structure theory of rings and fields
is definitive.  (Not the commutative ring stuff though!)  I bought it
before I really knew whether it was worth having; now I'm not sure,
but it's come in handy at surprising times.  Of dubious use as a
reference, since each chapter is woven rather tightly and he
frequently refers to hard results from volume I.

<A NAME="a:group-theory"><H2>Group theory and representations</H2></A>

<H4>Alperin/Bell, <i>Groups and representations</i></H4>

<P>If you're not into finite groups or their representations, this
book contains exactly what you need to know about them.  After a quick
run-through of what you probably already know, it treats matrix groups
(Alperin, like Artin, insists that these are the real examples of
finite groups, and I agree), p-groups, composition series, and then
basic representation theory via Wedderburn's structure theorem for
semisimple algebras.  I learned a lot from the matrix-groups chapter.
The exposition is nearly as clean and clear as Rudin's, and there are
many good exercises (some deliberately too hard, and none marked for
difficulty).

<P><b>[PC]</b> Yep, a solid text for an intro course to group theory
(at the graduate level).  It's designed so that no more and no less
than the entire book gets covered in Math 325, so unlike most math
books, I have read this from cover to cover.

<H4>Rotman, <i>Introduction to the theory of groups</i></H4>

<P>This is a group theorist's group theory book, although it contains
no representation theory at all.  What I've seen of it looks good (the
diagrams on the inside covers are neat, although I have no idea what
they mean).  But I don't like group theory that much, so I can't say
more.

<P><b>[BR]</b> This was my favorite reference for Murthy's 257
class. Starting with the simplest notions of permutations, Rotman is
able to construct everything you ever wanted to know about group
theory. If you're just looking for a clear, readable exposition and
elegant proofs of the isomorphism theorems or Sylow's theorems, this
is a great place to look.  And if by some random chance you have need
to learn what a wreath product is, you won't need to buy a new book.

<H4>Gorenstein, <i>Finite groups</i></H4>

<P><b>[BB]</b> The final word on finite groups prior to 1970.
Everything is in here.  Very hard reading for a non-specialist, but a
good reference for a serious group-theorist.  I think Glauberman has
it memorized.

<H4>Humphreys, <i>Introduction to Lie algebras and representation
theory</i></H4>

<P>A skinny little book which runs briskly through the basic theorems
on Lie algebras and their representations.  Note that it says Lie
algebras, not Lie groups; there are no smooth manifolds here!  There
are four copies in Eckhart Library and they're always all checked out,
so it must be pretty good; it helps that the alternative works (like
Jacobson, <i>Lie algebras</i>) are all very old, thus hard to read.

<H4>Fulton/Harris, <i>Representation theory: a first course</i></H4>

<P>This is a beautifully concrete introduction to Lie groups and their
representations.  &ldquo;First course&rdquo; in Joe Harris-speak means
that the book is driven largely by examination of concrete examples
and their characteristics: in fact, the first quarter of the book
covers representations of finite groups, as an extended
&ldquo;concrete example&rdquo; motivating the Lie theory.
Nevertheless the book is not easy reading, and you will need a lot of
multilinear algebra and some readiness to fill in glossed-over
details.  But at the end, you will know a lot about why the more
advanced general theory behaves as it does.  Physicists with a high
mathematics tolerance ought to check this one out.

<A NAME="a:ring-theory"><H2>Ring theory</H2></A>

<H4>Kaplansky, <i>Fields and rings</i></H4>

<P>Actually this is three little sheaves (coherent sheaves, even) of
lecture notes, bound as a book: one on Galois theory, one on the
classical structure theory of (noncommutative) rings, and one on
homological dimension theory of rings.  Kaplansky's exposition is
classic, and for people who (like me) didn't really get Galois theory
out of 259, this isn't a bad place to learn it.  He has a similar
volume called <i>Lie algebras and locally compact groups</i>, which is
half structure theory of Lie algebras and half (of all things) a proof
that a locally compact topological group has a unique analytic Lie
group structure.

<H4>Anderson/Fuller, <i>Rings and categories of modules</i></H4>

<P>Noncommutative rings have a homological theory very different in
flavor from that of commutative rings, namely the structure theory of
the categories <b>R-mod</b> and <b>mod-R</b> of left and right
modules.  I don't really know why I bought this book, because I find
the material itself pretty boring.  But it's a good exposition,
contains category-oriented proofs of most of the classical
noncommutative ring theory (as opposed to Lam's book below), and I did
use it to give a Math Club talk last year.

<H4>Morandi, <i>Field and Galois theory</i></H4>

<P>This is an exceedingly gentle but comprehensive course in field
theory (a lot more material than the field-theory chapter of a
general algebra text).  Morandi goes very slowly, and you could
probably cover most of the proofs and do them yourself; the beginning
exercises are too easy, but there are some good ones too.  You might
not find the material interesting enough to sustain such length of
presentation; if so, look at Kaplansky instead.  But it's a good
reference if you just need field theory to do something else with
(commutative algebra, say).

<H4>Lam, <i>A first course in noncommutative rings</i></H4>

<P>This is the ring-theory book I should have gotten when I was
looking at ring-theory books.  Informed by a huge number of examples
(many of which I never would have guessed could exist), Lam lays out a
beautiful and detailed exposition of the more concrete parts of the
theory of noncommutative rings as it exists today.  (Some more
sophisticated areas, such as the theory of central simple algebras
which Jacobson treats in <i>Basic algebra</i> II, are left to a
planned second course, now published as <i>Lectures on rings and
modules</i>.)  Lots of exercises, mostly not too hard.  He avoids
category-theoretic methods for the most part, which saves the book
from turning into the kind of functor catalog that Anderson/Fuller
sometimes becomes.

<A NAME="a:commutative-homological"><H2>Commutative and homological
algebra</H2></A>

<H4>Atiyah/Macdonald, <i>Introduction to commutative algebra</i></H4>
<H4>Matsumura, <i>Commutative ring theory</i></H4>
<H4>Eisenbud, <i>Commutative algebra with a view toward algebraic
geometry</i></H4>

As Pete Clark said, these three are the standard references now, in
roughly increasing order of difficulty.  Atiyah/Macdonald is short, to
the point, and mostly non-homological.  Matsumura is the &ldquo;big
Rudin&rdquo; of commutative algebra: a clear systematic exposition
from first principles.  Eisenbud is a huge, sprawling monster of a
book, which includes almost everything... somewhere.  All three have
many good exercises, and they complement each other well.  Eisenbud is
the newest and the most complete reference (and, as a specific
objective, includes every result used in Hartshorne's algebraic
geometry book), but it can be difficult to wade through so much
material to find what you want.  Atiyah/Macdonald is probably the best
introductory text&mdash;or try Kaplansky's book below.

<H4>Kaplansky, <i>Commutative rings</i></H4>

<P>I list this one separately because it's, well, different.  Like
Atiyah/Macdonald, this is a small book which takes up commutative
algebra from the beginning, largely without homological methods.
However, the pace is much brisker, and many results are stated in
somewhat idiosyncratic form, since Kaplansky resolutely avoids
algebraic-geometric language.  He unfortunately refers to the third
part of his notes <i>Fields and rings</i> (above) for the homological
results he does need.

<H4>Weibel, <i>An introduction to homological algebra</i></H4>

<P>Without this book I would probably have failed the second half of
Kottwitz's Math 327 class.  The first half is a systematic exposition
of homological algebra, more modern than the standard references: the
aim stated is to bring &ldquo;current technology&rdquo; in homological
algebra to casual users from other disciplines.  The second half is
devoted to a group of applications, including cohomology of groups
(the lifesaver in 327), Lie algebra homology and cohomology, and other
stuff.  It's reasonably well written and careful in notation (a very
important thing in this field).  Weibel also takes care not to let too
much abstract nonsense go by without an example or three of what in
the hell structures he might be talking about.

<A NAME="a:number-theory"><H2>Number theory</H2></A>

<H4>Weil, <i>Basic number theory</i></H4>

<P><b>[PC]</b> Um, I saw this book in the Coop, was intrigued by the
title, and opened it up to a discussion of Haar measure!  Not suitable
for a first course in number theory, or a second course in number
theory, or... It's really hard.  Maybe someday I'll get to it.

<P><b>[CJ]</b> It's not <i>that</i> bad, just... brisk.  Weil was
another of the original Bourbakistes, and his approach to algebraic
number theory reflects their devotion to proper foundation: to study
global (algebraic number) fields, one must first study local (locally
compact) fields, and to study these one begins with topology and
measure, etc.  I think it's a great book, but it's true you won't
learn any number theory you don't already know.  You'll discover that
you hadn't known what you thought you knew, but now you do.

<H4>Narkiewicz, <i>Introduction to the elementary and analytic theory
of algebraic numbers</i></H4>

<P>This is a huge yellow brick which looks more like a dictionary than
a math book.  Narkiewicz gives a careful exposition of basic algebraic
number theory (in somewhat old-fashioned notation) with more emphasis
on the role of (both complex and p-adic) analytic methods than usual.
I used it to learn some things about character theory on the p-adics.
Notable for its extensive historical notes, unsolved problems lists,
and truly immense bibliography.

<H4>Silverman, <i>The arithmetic of elliptic curves</i></H4>

<P>Silverman's two books (the second is <i>Advanced topics in the
arithmetic of elliptic curves</i>) are the standard texts in the
subject, and from what I've seen they deserve it.  You will need to be
thoroughly comfortable with basic algebra and number theory to pick up
the first one, however.  If you want to learn something about elliptic
curves without so much algebraic background, try Koblitz,
<i>Introduction to elliptic curves and modular forms</i> (but brush up
your complex analysis) or Cassels, <i>Lectures on elliptic curves</i>
(and be prepared for a short book that doesn't hold your hand much).

<H4>Koblitz, <i>p-adic numbers, p-adic analysis, and zeta
functions</i></H4>

<P><b>[PC]</b> Interesting, and probably a good place to read up on
p-adics.

<P><b>[CJ]</b> I still want to know what a zeta function really is.
Koblitz is a good writer, and he'd probably tell me if I read his
book...

<H4>Fr&ouml;hlich/Taylor, <i>Algebraic number theory</i></H4>

<P><b>[PC]</b> This is the book that I'd love to find time to read
from cover to cover.  It's advanced in the sense that it's definitely
for would-be algebraic number theorists: they cover a lot of ground
and basically pride themselves on doing stuff that the other
introductory texts don't. For example, they actually talk about cubic,
biquadratic and sextic number fields, and complain in their
introduction that many number theorists never acquire enough technique
to work with anything but quadratic fields.  But in terms of
prerequisites, it presupposes a <i>solid</i> knowledge of
undergraduate algebra, including an acquaintance with modules.  I'm
biased because I love algebraic number theory, but this book jumped
onto my shelf above all the others.  There is just so much great stuff
in here, and it is written about with enthusiasm and clarity.  Only
problem is the confusing and oppressive letters that they use for
ideals; what's up with that?

<P><b>[CJ]</b> What, the lower-case Fraktur?  It's the old standard
(grin).

<A NAME="a:combinatorics"><H2>Combinatorics and discrete
mathematics</H2></A>

<H4>Lovasz, <i>Problems in combinatorics</i></H4>

<P><b>[PS]</b> You simply must include what Hungarian mathematicians
consider the most important math book ever, Laszlo Lovasz's huge tome
covering combinatorics from an elementary level to Ph.D. level in one
book.  It teaches combinatorics the way Hungarians think it should be
taught, by doing lots of problems.  The problems are very hard, but in
the book there are separate sections for problems, hints (which are
often quite helpful), and full solutions.  Every budding young
Hungarian combinatorist spends a year doing every problem in this book
sometime before he finishes his Ph.D.  As a side treat, the questions
are often filled with bits of Hungarian culture, e.g. &ldquo;How many
ways can you pass out <i>k</i> forints to <i>n</i> friends if 1 friend
only wants an even number of forints and the rest of them must get at
least one?&rdquo; or &ldquo;Bela wants to buy flowers for his
friend...&rdquo; Probably the main thing wrong with this book is it's
horribly expensive unless you buy it in Hungary, where it's still $60.
If you can't find this book in Eckhart, then maybe it's not so
important to include it.  On the other hand, Babai did help write it,
so it is relevant nonetheless.

<P><b>[CJ]</b> A forint is about half a cent these days.

<H4>Stanley, <i>Enumerative combinatorics</i> I</H4>

<P>Combinatorics is maturing from a collection of problems knit
together by ad hoc methods (or methods which appear ad hoc to
non-combinatorists) into a discipline which is taught and learned
systematically.  Stanley's book got a rave review in the Bulletin of
the AMS as the new standard reference on counting, which really means
most of combinatorics; I haven't read it but I've seen it on a whole
lot of grad students' shelves.  Try it out if G/K/P (above) is too
talky for you.  The second volume is now out.

<H4>Bollob&aacute;s, <i>Modern graph theory</i></H4>

<P>This recent Springer GTM is a substantial revision and expansion of
Bollob&aacute;s's earlier graph theory text.  Although I'm not a
combinatorist by any stretch of the imagination, it looks like a good
book, inviting but not toy.

<A NAME="a:measure-theory"><H2>Measure theory</H2></A>

<H4>Halmos, <i>Measure theory</i></H4>

<P>This was the standard reference for at least two generations of
analysts, and it probably still is, because nobody writes books
entitled <i>Measure theory</i> any more.  Basically it's an abstract
analysis text with extra care paid to set-theoretic questions,
regularity problems for measures, and a construction of Haar measure.
It's a good book, since Paul Halmos wrote it, but it might be
considered old-fashioned now.  (For a more modern, emphatically
measure-theoretic analysis text, check out Bruckner/Bruckner/Thomson,
<i>Real analysis</i>.)

<H4>Federer, <i>Geometric measure theory</i></H4>

<P>Federer's book is listed here because in the last few months, to my
great surprise, it has become my reference of choice for basic real
analysis (replacing the first half of big Rudin).  Chapter 2 (of 5) is
entitled &ldquo;General measure theory&rdquo;, and it covers chapters
1&ndash;3 and 6&ndash;8 of big Rudin in the space of eighty pages,
together with tons of additional material on group-invariant measures,
covering theorems, and all the geometric measures (Hausdorff et al).
The presentation is compressed to within epsilon of unreadability, but
once you unravel it, it has a powerful elegance.  Federer takes great
care to give the limits of generality in which each result is true.
There are no exercises, but reading the book is hard exercise enough.
My one quibble is that even big-name theorems are referenced by
number; I would far prefer &ldquo;by the dominated convergence
theorem&rdquo; to &ldquo;by 2.3.13&rdquo; for the rest of the book.
If you don't like reading dense books, stay far, far away from
Federer, but if you want a complete, powerful reference to measure
theory, give it a try.

<A NAME="a:probability"><H2>Probability</H2></A>

<H4>Feller, <i>Introduction to probability theory and its
applications</i></H4>

<P>This is the standard text.  It splits into two volumes, namely
probability before and after it turns into measure theory.  What I've
read of it is quite well written, and noteworthy for the great care
with which it discusses experimental issues (the idea &ldquo;what
sequence of choices corresponds to what mathematical construct&rdquo;
can get sticky when dependence relations are complex).  Some of us
will need to know some probability someday, and here it is.
Alternative references are Shiryaev, <i>Probability</i> (Springer, so
cheaper and easier to get, but very Russian) and Billingsley,
<i>Probability and measure</i> (by a UC emeritus).

<A NAME="a:functional-analysis"><H2>Functional analysis</H2></A>

<H4>Conway, <i>A course in functional analysis</i></H4>

<P>A grad student I knew from 325 saw me leaving the bookstore with
this book, and told me it was terrible, that he'd hated it at
Dartmouth.  I didn't believe him at the time, but now I see what he
meant.  As in his complex analysis book, Conway develops functional
analysis slowly and carefully, without excessive generalization
(locally convex spaces are a side topic) and with proofs in great
detail, except for the ones he omits.  This time around, though, the
detail is excruciating (many functional analysis proofs consist of a
mass of boring calculation surrounding one main idea) and the notation
is simply awful.  (The fact that Hilbert spaces are often function
spaces is <i>not</i> an excuse to use <i>&lsquo;f&rsquo;</i> to denote
a general element of a Hilbert space.)  The book is not without
virtues, but it goes so slowly that I can't see which results are
important.

<H4>Dunford/Schwartz, <i>Linear operators</i></H4>

<P>After all these years, I think Dunford/Schwartz is still the bible
of functional analysis; the analysts who did all the exercises in
Kelley to learn topology tried to do all the exercises in here, or at
least volume 1, to learn about operators.  They all failed, although
one of the exercises turned into Langlands's doctoral thesis.  D/S is
too old to be easily read now, but worth looking at for culture.

<H4>Kadison/Ringrose, <i>Fundamentals of the theory of operator
algebras</i></H4>

<P>No, I'm not turning into an operator algebraist (although I might
be doing noncommutative geometry some day).  The first three-fifths of
volume 1 contains a much better treatment of basic functional analysis
than I've seen elsewhere, certainly slanted toward operator algebras,
but clearly written and <i>interesting</i> (a quality lacking in many
functional analysis texts).  The book is known for its collection of
challenging exercises, which were so popular that K/R wrote up
complete solutions to the two volumes and published them as volumes 3
and 4.  Unfortunately volume 1 is missing from Eckhart Library.

<H4>Kreyszig, <i>Introductory functional analysis with
applications</i></H4>

<P>Here is a book to look at for a lot of applications and motivation
for functional analysis, without a lot of technicalities.  I've only
looked at it a little bit; it seems to be written more like a physics
book, substituting a plausibility argument for an occasional tricky
technical proof, but spending a lot of time in explanation.  Try it if
you have trouble seeing what's really different about the
infinite-dimensional case.

<H4>Zimmer, <i>Essential Results of Functional Analysis</i></H4>

<P><b>[BB]</b> It's a U of C published blue book, and is extremely
concise and quickly presents most of the stuff one needs to know.
It's certainly not easy&mdash;Chapter 0 presents weak
derivatives&mdash;but it's a good second course.

<A NAME="a:complex-analysis"><H2>Complex analysis</H2></A>

<H4>Andersson, <i>Topics in complex analysis</i></H4>

<P>I got through the non-Riemann surfaces part of 314 on this book.
It's a skinny Springer Universitext which presents complex analysis at
a second-course level, efficiently and clearly, with less talk and
fewer commercials.  He starts off by defining <i>dz = dx + i dy</i>,
which will annoy some people but makes me happy.  Later chapters treat
more advanced analytic material (Hardy spaces, bounded mean
oscillation, and the like).  The exercises are pretty tough.

<H4>Gunning/Rossi, <i>Analytic functions of several variables</i></H4>

<P>This is one of the classic texts on the &ldquo;real&rdquo; theory
of several complex variables, meaning analytic spaces, coherent
sheaves and the whole bit.  It's a good book so far as it goes, but
there's a lot of hard theory and not a lot of geometric
motivation&mdash;and no exercises.

<H4>Whitney, <i>Complex analytic varieties</i></H4>

<P>And this is where you go to learn the &ldquo;fake&rdquo; theory of
several complex variables, meaning what things actually look like
geometrically, with as little machinery as possible.  Very concrete.
I think there's a law that several-complex-variables books must have
no exercises and must use letters as ordinals at some sectioning
level.

<H4>Narasimhan, <i>Compact Riemann surfaces</i></H4>

<P>I put this book here to warn that, although Corlette likes to use
it as a 314 text, you should not try to read it until your second or
third year of graduate school.  It presents the theory of compact
Riemann surfaces as someone who already knew the general principles
would see it, as a specialization of complex algebraic geometry.

<P><b>[PC]</b> This book lies on my shelf from Math 314, waiting for
someone smarter than me to come by and read it.  I think I read pages
27 and 28 about 50 times, but that's about it.

<H4>Jost, <i>Compact Riemann surfaces</i></H4>

<P>If you want to know what Riemann surfaces are and why they're
interesting, go here instead.  Jost assumes little background; you
could probably read this after 207-8-9 with some work.

<H4>Weyl, <i>The concept of a Riemann surface</i></H4>

<P>Or try this book, which is a beautiful classic but uses terminology
and ways of thinking which we consider archaic.  Hassler Whitney is
credited with the formal definition of a differentiable manifold, and
Riemann with the idea (in his <i>Habilitationsschrift</i>; see Spivak
volume 2 for a translation), but the first edition of this book was a
significant step in its formulation.  Read for culture and brain
elevation, once you know some substantial complex analysis.

<A NAME="a:harmonic-analysis"><H2>Harmonic analysis</H2></A>

<H4>Katznelson, <i>An introduction to harmonic analysis</i></H4>

<P>And he means <i>analysis</i>... This is a short text on classical
harmonic analysis, cheap and pretty readable.  There's a rather
perfunctory treatment of locally compact groups at the end, but the
real emphasis is on the classical theory of Fourier series and
integrals, including all kinds of sticky convergence and summation
questions.

<H4>Rudin, <i>Fourier analysis on groups</i></H4>

<P>This is a classic text on commutative harmonic analysis (that is,
on locally compact <i>abelian</i> groups).  It's a fairly dense
research monograph.

<H4>Hewitt/Ross, <i>Abstract harmonic analysis</i></H4>

<P>H/R is the Dunford/Schwartz of harmonic analysis; this is an
immense two-volume set which spends most of a first volume just
setting up the generalities on topological groups and integration
theory.  As such, the recommendation is similar: look at it for
culture.

<H4>Stein/Weiss, <i>Introduction to Fourier analysis on Euclidean
spaces</i></H4>

<P>You might think of this as a more advanced Katznelson; it requires
a pretty solid comfort with first-year graduate analysis to read.

<H4>Helgason, <i>Groups and geometric analysis</i></H4>

<P>I found this a fascinating book.  At the risk of totally missing
the point I might characterize it as the differential-geometric side
of noncommutative harmonic analysis (infinite-dimensional
representation theory of nonabelian groups).  It's about the geometric
objects which arise from invariance under symmetries of an ambient
space (e.g., the Laplacian is the only isometry-invariant differential
operator on the plane).  Maybe someday I will actually be able to read
it; Helgason's earlier book (below) is a sufficient preparation.

<A NAME="a:differential-equations"><H2>Differential equations</H2></A>

<H4>Taylor, <i>Partial differential equations I: basic theory</i></H4>

<P>I finally learned a little about PDEs, and this book is the first
one I'd recommend to any pure mathematicians interested.  It's the
first volume of a monumental three-volume series covering a wide range
of topics in analysis and geometry (yes, Atiyah-Singer is in volume
II).  Volume I contains the foundational material on Fourier analysis,
distributions and Sobolev spaces, application to the classical
second-order PDE (Laplace, heat, wave, et cetera), as well as a handy
introductory chapter containing all you really need to know about
ordinary differential equations!  This list of topics doesn't do the
book justice, however, since it's <i>packed</i> with interesting
little applications and side notes, in the text and the copious
exercises.  The general consensus among MIT graduate students is that
this book, like Federer and Griffiths/Harris, has everything in the
world in it.

<H4>Evans, <i>Partial differential equations</i></H4>

<P>This is a big, fat, talky introduction to PDE for pure
mathematicians.  It slights some theoretical topics (Fourier
transforms and distributions) in favor of an unusually full treatment
of nonlinear PDE; the author claims that &ldquo;we know too much about
linear equations and not enough about nonlinear ones,&rdquo; and his
preferences are evident throughout.  But it is a good book, written
with careful attention to pedagogy and making things make sense to
someone new to the field.  I like it as a textbook, but Taylor is a
better first choice for reference.

<H4>H&ouml;rmander, <i>The analysis of linear partial differential
operators</i> I</H4>

<P>Here is the book Evans was complaining about; H&ouml;rmander's
four-volume masterwork contains everything we knew about linear PDE up
to the mid-seventies.  The first volume is available as a paperback
study edition, and makes a good secondary reference on distributions
and Fourier transforms.  I hope someday to understand the last two
chapters, which introduce something called &ldquo;microlocal
analysis&rdquo; that currently has me fascinated.  The book shows
little mercy for the reader; distribution theory has some very hard
technicalities and H&ouml;rmander proceeds pretty briskly.  But it's
sometimes nice to have a truly definitive reference.

<H4>Olver, <i>Equivalence, invariants and symmetry</i></H4>

<P>Another book on geometric objects arising from invariance
conditions, this one more focused on differential equations.  People
confused about why the equations of physics look the way they do might
try it.

<A NAME="a:differential-topology"><H2>Differential topology</H2></A>

<H4>Hirsch, <i>Differential topology</i></H4>

<P><b>[PC]</b> A solid introduction to differential topology, but
maybe a bit bogged down in technical details: a theme of the subject
is that arbitrary maps can be approximated by very nice maps under the
right conditions.  Hirsch has a chapter which he investigates
conditions <i>other</i> than &ldquo;the right ones,&rdquo; and comes
up with some sharpish estimates about when you can approximate what by
what.  This is sort of interesting, but seems distinctly antithetical
to the spirit of &ldquo;soft&rdquo; analysis which runs through my
veins and the veins of differential topologists everywhere.  Why
bother?  I own the book, and there's some good stuff in it, but in
retrospect I'd rather own Guillemin and Pollack, which proceeds a bit
more geometrically and far less rigorously.  The rigor is optional and
can be filled in later.

<P><b>[CJ]</b> I agree with Pete's assessment of the book, but not
with his opinions on rigor.  Hirsch is a good second differential
topology book; after you see how all the touchy-feely stuff goes (move
it a little bit to make it transverse), read Hirsch to see how it
actually works, and how a nice theoretical framework can be
constructed around the soft geometric ideas.  I think it's
indispensable to see how things are <i>done</i>.

<H4>Lang, <i>Differential and Riemannian manifolds</i></H4>

<P>Another Serge Lang book, which <i>also</i> contains a proof of the
inverse function theorem in Banach spaces (sigh).  It's not really
human-readable, and I list it mostly because it was the first
manifolds book I blundered across in 209.  But it has a nice proof of
the ODE existence theorem, too.

<H4>Warner, <i>Foundations of differentiable manifolds and Lie
groups</i></H4>

<P>This is a curious selection of material: besides the basic theory
of manifolds and differential forms, there is a long chapter on Lie
groups, a proof of de Rham's theorem on the equivalence of de Rham
cohomology to Cech and topological cohomology theories, and a proof of
the Hodge theorem for Riemannian manifolds.  It's convenient to have
all this stuff here in a single book, but Warner's notation annoys me
terribly, and you can find better treatments of any one topic
elsewhere.

<A NAME="a:algebraic-topology"><H2>Algebraic topology</H2></A>

<H4>Massey, <i>A basic course in algebraic topology</i></H4>

<P>Massey wrote two earlier algebraic topology books, <i>Algebraic
topology: a first course</i>, and <i>Singular homology theory</i>.
This book is their union, minus the last chapter or two of the first
book.  Thus the first half of the book is a nice, well-grounded
treatment of the fundamental group and covering spaces, at a very
elementary level (Massey fills in all the material on free groups and
free products of groups).  The second half is a course on homology
theory which is, well, boring.  Too slow, too elementary, too talky,
and not even very geometric for all that.  It'll do, but it's not
lovable.

<P><b>[PC]</b> For better or worse, this will probably be your first
textbook on algebraic topology.  I know Chris doesn't like it very
much.  The homotopy theory part is fine, but I think the
homology/cohomology part could be improved... somehow.

<H4>Fulton, <i>Algebraic topology: a first course</i></H4>

<P><b>[PC]</b> I own this too, and it's a pleasant book: an algebraic
topology book for math students who aren't especially interested in
algebraic topology.  No, really.  I do like algebraic topology, but
this book appeals to me too because it takes a holistic and geometric
approach to the material; after all, algebraic topology is supposed to
be for proving stuff about manifolds and complexes (and other
topological spaces of interest, if any), not about chain complexes.
There's a lot of interesting stuff here, but because Fulton often
contents himself with &ldquo;the simplest nontrivial case&rdquo; for
fundamental groups, homology, etc., the presentation is less than
complete.  Great supplementary reading and good treatment of branched
covering spaces.

<H4>Bott/Tu, <i>Differential forms in algebraic topology</i></H4>

<P>This book made algebraic topology make sense to me!  Bott/Tu
approach cohomology and homotopy theory through the de Rham complex,
which means the calculations are all easy to understand and give
insight into the geometric situation.  The book is not a first course
in algebraic topology, as it doesn't cover nearly all the standard
topics.  What it does cover is beautifully clear, motivated and, well,
sensical.  They even give a good excuse for spectral sequences, which
in my book is a major accomplishment.

<H4>Spanier, <i>Algebraic topology</i></H4>

<P>Spanier is the maximally unreadable book on algebraic topology.
It's bursting with an unbelievable amount of material, all stated in
the greatest possible generality and naturality, with the least
possible motivation and explanation.  But it's awe-inspiring, and
every so often forms a useful reference.  I'm glad I have it, but most
people regret ever opening it.

<H4>Rotman, <i>An introduction to algebraic topology</i></H4>

<P><b>[BR]</b> You didn't mention this one.  I think an appropriate
nickname for this one is &ldquo;Spanier Lite&rdquo; or maybe
&ldquo;Diet Spanier&rdquo;, or better still, &ldquo;Spanier for
Dummies.&rdquo; Rotman was actually a student of the infamous Spanier
(and also of Saunders Mac Lane for that matter!).  Basically, he stole
the table of contents from Spanier's book and tried to write a text
that was much less dense and general, but more in depth and more
categorical than, say, Massey.  I've only read through the first 3
chapters, but anyone who is totally frustrated with having to choose
between ultra-elementary and ultra-advanced algebraic topology books
should look here.

<H4>Stillwell, <i>Classical topology and combinatorial group
theory</i></H4>

<P><b>[PC]</b> This book is great!  No book on this list coincides
with my own mathematical esthetics like this one: I checked this book
out this summer while I was doing research on surface topology and
read it cover to cover: you'll see how geometry relates to topology
relates to group theory.  I wish this was my first algebraic topology
book, because it's full of exciting theorems about surfaces,
three-manifolds, knots, simple loops, geodesics&mdash;in other words,
it's rippling with geometric/topological content intead of commutative
diagrams.  Let me also recommend Stillwell's book <i>Geometry of
surfaces</i>, along the same lines.

<H4>Bredon, <i>Topology and geometry</i></H4>

<P>Don't be fooled by the word &ldquo;geometry&rdquo; in the title;
there are two chapters on basic differential topology followed by the
best modern course in basic algebraic topology I've seen.
Differential geometry and Lie groups supply the occasional example,
but there are no metrics to be found!  Lots and lots of exercises.

<P><b>[PC]</b> This one gets the Ben Blander seal of approval.  From
what I've seen, it's an excellent compendium of graduate-level
geometry and topology powered by good examples and (again!) actual
geometric content.

<A NAME="a:differential-geometry"><H2>Differential geometry</H2></A>

<H4>Spivak, <i>A comprehensive introduction to differential
geometry</i>, 3-5</H4>

<P>The latter three volumes form the &lsquo;Topics&rsquo; section of
Spivak's masterwork; he treats a succession of more advanced theories
within differential geometry, with his customary flair and the
occasional stop for generalities.  The last chapter is entitled
&ldquo;The generalized Gauss-Bonnet theorem and what it means for
mankind&rdquo;, so that gives you an idea of Spivak's take on
geometry.  Sadly again, there are no exercises, but the annotated
bibliography at the end of volume 5 is immense.

<H4>Helgason, <i>Differential geometry, Lie groups, and symmetric
spaces</i></H4>

<P>The title is a little bit of a misnomer, as this book is really
about the differential geometry <i>of</i> Lie groups and symmetric
spaces, with an occasional necessary stop for Lie algebra theory.  The
first chapter is a rapid if rather old-fashioned (no bundles; tensors
are modules over the ring of smooth functions) course in basic
differential geometry.  The rest of the book describes the geometric
properties of symmetric spaces (roughly, manifolds with an involutive
isometry at each point) in depth.  I find the material interesting in
itself, and as a lead-in to Helgason's other fascinating book (above).
There are many exercises, and <i>solutions</i> at the end!

<H4>Kobayashi/Nomizu, <i>Foundations of differential geometry</i></H4>

<P>K/N is the standard reference on differential geometry from the
sophisticated point of view of frame bundles.  The emphasis here is on
&lsquo;reference&rsquo;, unfortunately.  I think it's the only book
anyone actually uses to look up stuff about principal bundles when
they need it, but it's not written as a textbook.  The notes and
bibliography are very nice, however.

<H4>Rosenberg, <i>The Laplacian on a Riemannian manifold</i></H4>

<P><b>[BB]</b> A different approach to geometry, through analysis.
Lots of exercises integrated critically into the text; proves the
Hodge theorem using the heat kernel.  Introduces analysis on
manifolds.  I've only gotten through the first chapter and I've
skimmed the rest, so I can't say too much more, but it looks
interesting.

<H4>do Carmo, <i>Riemannian geometry</i></H4>

<P><b>[BB]</b> A readable and interesting introduction to the subject.
It covers some interesting material, such as the sphere theorem and
Preissman's theorem about fundamental groups of manifolds of negative
curvature, and much more.

<H4>Boothby, <i>Introduction to differentiable manifolds and
Riemannian geometry</i></H4>

<P>I don't know why everyone likes this book so much; maybe because
they managed to find it and it contains what they need?  It's just
another manifolds book, really, and less well-written (lots of
annoying coordinates) than most.

<A NAME="a:geometric-measure-theory"><H2>Geometric measure
theory</H2></A>

<H4>Morgan, <i>Geometric measure theory: a beginner's guide</i></H4>
<H4>Mattila, <i>Geometry of sets and measures on Euclidean
spaces</i></H4>
<H4>Federer, <i>Geometric measure theory</i></H4>

<P>Okay, so it's a little overkill, but I <i>like</i> geometric
measure theory.  Here are three books about it, two you should
consider reading and one you should consider <i>not</i> reading.
Morgan truly is a beginner's guide, and one of the best I've seen to
any subject.  He introduces the formidable technical apparatus of
geometric measure theory bit by bit, leaning on pictures and examples
to show what it's for and why we work so hard.  Proofs of hard
theorems are frequently omitted (mostly referred to Federer).  Mattila
is a recent book on the theory of rectifiability, and looks good from
the little I've seen.  Federer is the bible, and it's the densest book
I've ever seen, on anything.  Everything up to 1969 is in here, and
much afterward is anticipated.  In addition to the theory of
rectifiable sets, Federer develops a powerful homological integration
theory, leading to a homology theory for locally Lipschitz sets and
maps in <b>R</b><sup>n</sup> which is isomorphic on nice sets to the usual
homology theories.  You can't really learn from it, except that
sometimes you have to: the subject is itself very complicated and
there are few expositions.

<H4>Falconer, <i>The geometry of fractal sets</i></H4>

<P>Here is an exposition of the rudiments of geometric measure theory,
mostly Hausdorff measures, together with applications to
rectifiability and regularity of sets of ugly dimension.  A nice
little book if you're curious about why it's a cool subject.

<A NAME="a:algebraic-geometry"><H2>Geometry: algebraic
geometry</H2></A>

<H4>Harris, <i>Algebraic geometry: a first course</i></H4>

<P>Algebraic geometry is a hard subject to learn, and here is as good
a place as any.  It has a very different flavor from any other kind of
geometry we study in this day and age: lots of results about curves
having cusps and intersecting hyperplanes three times.  Harris
presents a body of classical material (projective varieties over an
algebraically closed field of characteristic zero) through analysis of
many, many examples, much like his representation theory book.  Be
warned that much is left out, and you develop your first familiarity
with the subject by figuring out what he's really saying.  You will
also need to be quite comfortable with multilinear algebra.  But
Harris has a great expository style, and there's a lot of good stuff
in all those examples.

<H4>Shafarevich, <i>Basic algebraic geometry</i> 1</H4>

This may be a better place to learn for the first time, as Shafarevich
assumes that the language and ways of thought of algebraic geometry
are alien to the reader.  He proceeds briskly, though, with fewer
stops to look around for interesting examples of varieties
(ameliorated somewhat by the copious exercises).  To make a serious
attempt at learning algebraic geometry, you'll probably need both.
Shafarevich, like Harris, teaches some of the commutative algebra
along the way.

<H4>Mumford, <i>Algebraic geometry I: complex projective
varieties</i></H4>

<P>This book is superficially similar to the previous two (varieties,
no schemes) but it's written for mature mathematicians: it's an
expository monograph, not a textbook.  As such, it's a Good Book in
the abstract, but not all that useful to someone looking for guidance.
You will need to be solidly comfortable with commutative algebra to
begin reading.

<H4>Griffiths/Harris, <i>Principles of algebraic geometry</i></H4>

<P>A huge, sprawling, beautiful, inspiring, infuriating book.  It
should be called <i>Principles of analytic geometry</i>, because
although the questions are algebraic-geometric, the objects and
methods considered are all complex-analytic.  This is algebraic
geometry over <b>C</b>, the classical case and the one in which
existing theory is richest.  It's a beautiful and hugely sophisticated
theory.  G/H treat a vast quantity of it in eight hundred pages, and
the treatment is still so compressed that many proofs are quite
elliptical.  Filling in the gaps requires (or develops) a great deal
of maturity.  If you're interested in any aspect of algebraic or
differential geometry, you should not miss this book&mdash;but don't
expect any of it to be easy.

<H4>Hartshorne, <i>Algebraic geometry</i></H4>

<P>Hugh, my algebra TA, described Hartshorne as &ldquo;the schemes
book for the more manly algebraic geometer&rdquo;.  It's the standard
exposition of scheme theory, the Grothendieck remaking of algebraic
geometry, and it's legendarily difficult, not only the text but the
many exercises.  The preface to Shafarevich's English edition remarks
that &ldquo;many graduate students (by no means all) can work very
hard on Chapters Two and Three of Hartshorne for a year or more, and
still know more or less nothing at the end of it.&rdquo; But, as with
most legendarily difficult books, it has its own awesome beauty, and
the diligent reader is rewarded.  I'm not sure Hartshorne belongs in
an undergraduate bibliography, but I did say &ldquo;difficulty level
unbounded above&rdquo;...

<P><HR>
Undergraduate mathematics bibliography, revised 21 January 1999 (142
entries)<BR>
Christopher Jeris, [first initial][last name] at fas.harvard.edu

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